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Chapter 4: Problem 46

Simplify. $$ (3 x-4 y)+(4 y-3 x) $$

Short Answer

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Step by step solution

01

Identify like terms

Group the like terms from the expression \( (3x - 4y) + (4y - 3x) \).
02

Combine like terms

Combine the like terms: \( 3x - 3x \) and \( -4y + 4y \).
03

Simplify

Simplify the expression by performing the arithmetic operations: \( 3x - 3x = 0 \) and \( -4y + 4y = 0 \).
04

Write the final answer

Combine the simplified terms: \( 0 + 0 = 0 \). Thus, the simplified expression is \( 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

combining like terms
When we simplify algebraic expressions, a vital skill is combining like terms. Like terms are parts of an expression that have the same variable raised to the same power. For example, in the expression \( 3x - 4y + 4y - 3x \), \(3x \) and \(- 3x\) are like terms, while \(-4y \) and \(4y\) are also like terms. Here's a simple way to identify and combine them:
  • First, group the like terms together.
  • Second, combine them by performing the arithmetic operation.
In our example, \(3x - 3x \) and \(-4y + 4y \) are grouped and then combined to get 0.
arithmetic operations
Arithmetic operations are the basic mathematical actions we perform when handling numbers and expressions. The primary operations are addition, subtraction, multiplication, and division. When simplifying algebraic expressions like \( (3x - 4y) + (4y - 3x) \), it's crucial to:
  • Correctly identify and align the numbers or terms involved.
  • Perform operations step-by-step.

    • For instance, in our example:

    • Adding \(4y \) to \(-4y\) yields 0.
    • And subtracting \(3x \) from \(3x\) also results in 0.
    Finally, you sum up the results, which gives us the simplified expression.
algebraic simplification
Algebraic simplification is the process of making an algebraic expression as easy to work with as possible. This often involves combining like terms and performing arithmetic operations. Simplifying the expression \( (3x - 4y) + (4y - 3x)\) shows a practical application of these steps. Here鈥檚 a clear path to achieving simplification:
  • Identify and arrange like terms.
  • Combine like terms by performing the necessary operations.
  • Sum the results to write the final simplified form.
For \((3x - 4y) + (4y - 3x) \), when you combine the positives and negatives of like terms, you are left with \(0\). This demonstrates how, through careful simplification, an expression can be broken down to its most fundamental form.

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Most popular questions from this chapter

Write a system of equations for each problem, and then solve the system. The two domestic top-grossing movies of 2009 were Avatar and Transformers 2: Revenge of the Fallen. The movie Transformers 2 grossed \(\$ 26.9\) million less than Avatar, and together the two films took in \(\$ 831.1\) million. How much did each of these movies earn?

Solve each equation for \(y\). $$ -2 x+y=9 $$

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Solve each system by the elimination method. $$ \begin{aligned} &5 x+4 y=12\\\ &3 x+5 y=15 \end{aligned} $$

Solve each system by the elimination method. $$ \begin{aligned} 5 x-4 y &=-1 \\ x+8 y &=-9 \end{aligned} $$
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